How can I Make Sure No Hair Is In The Mesh?
In your case the hair cannot get inside as the default Path render method is used. Shorten the hair as I can see it has the default length which is way too long. Make the length something like 0.1. And If you want hair all over your mesh, increase the hair number. If it's too low not all faces get it. Like check at the info bar how many faces your mesh has, multiply this number e.g. by 2 and input this value as the hair number. Set your Hair/face to 2 et voila.